Statistical Blending-Type Approximation by a Class of Operators That Includes Shape Parameters λ and α

Abstract

This paper is devoted to studying the statistical approximation properties of a sequence of univariate and bivariate blending-type Bernstein operators that includes shape parameters $\alpha$ and $\lambda$ and a positive integer. An estimate of the corresponding rates was obtained, and a Voronovskaja-type theorem is given by a weighted A-statistical convergence. A Korovkin-type theorem is provided for the univariate and bivariate cases of the blending-type operators. Moreover, the convergence behavior of the univariate and bivariate new blending basis and new blending operators are exhaustively demonstrated by computer graphics. The studied univariate and bivariate blending-type operators reduce to the well-known Bernstein operators in the literature for the special cases of shape parameters $\alpha$ and $\lambda ,$ and they propose better approximation results.

1. Introduction

The uniform approximation of continuous functions by polynomials was one of the problems that Karl Weierstrass focused on (see [1]). The classical Weierstrass approximation theorem asserts that there exists a sequence of polynomials $r_p\left(u\right)$ that converges uniformly to $r\left(u\right)$ for any continuous function $r\left(u\right)$ on the closed interval $[a,b]$. Later, Bernstein provided an alternative proof of the well-known Weierstrass approximation theorem currently called Bernstein polynomials (see [2]). The following Bernstein operators:

$${\mathcal{B}}_p(r;u)=\sum _{i=0}^p{b}_{p,i}\left(u\right)r\left(\frac{i}{p}\right),$$

where,

$$b_{p,i}\left(u\right)=\left(\genfrac{}{}{0pt}{}{p}{i}\right)u^i{(1-u)}^{p-i},u\in [0,1]$$

were given in [2] to approximate a given continuous function $r\left(u\right)$ on $[0,1].$

The Bernstein approximation technique has been used in both applied mathematics and functional analysis. For instance, an algorithm for the approximate solution of singularly perturbed Volterra integral equations and a numerical scheme for the computational solution of a new class of Volterra integral equations of the third kind via the Bernstein-type operators were proposed in [3,4].

Recently, Chen et al. constructed a new family of Bernstein operators for the continuous function $r\left(u\right)$ on $[0,1]$, which includes the shape parameter $\alpha$, and named it the $\alpha$-Bernstein operators [5]. They investigated certain elementary properties of these operators, such as end-point interpolation, linearity, and positivity, and obtained an upper bound for the error in terms of the usual modulus of continuity. Many variations of $\alpha$-Bernstein operators have been examined (see [6,7,8]).

A new basis with shape parameter $\lambda \in [-1,1]$ was introduced in [9] to give a practical algorithm of curve modeling. The authors studied certain important properties of the basis function and the related curves, and extended their research to the tensor product surface with two shape parameters. A new type of $\lambda$-Bernstein operators was constructed by shape parameter $\lambda$ in [10]. A Korovkin-type approximation theorem was provided; a local approximation theorem was investigated; a convergence theorem for the Lipschitz continuous functions was given; a Voronovskaja-type asymptotic formula was obtained as well.

Recently, shape parameters $\alpha$ and $\lambda$ were used to extend Bernstein operators to $\alpha$-Bernstein type (see [5,6,8,11,12,13,14,15,16,17,18]) and $\lambda$-Bernstein type operators (see [9,19,20,21,22,23,24,25,26]) in order to approximate functions better, respectively. The convergence of linear positive operators by the shape parameters $\alpha$ and $\lambda$ in the space of continuous functions of two variables was studied in [27,28,29] and [22,30,31], respectively. In [22], certain estimates of the differences of $\lambda$-Bernstein and $\lambda$-Durrmeyer and $\lambda$-Bernstein and $\lambda$-Kantorovich operators were established.

This paper is focused on the statistical approximation properties of univariate and bivariate $(\alpha ,\lambda ,s)$-Bernstein operators. The corresponding rates of convergence were estimated by the weighted A-statistical convergence. A Korovkin-type and a Voronovskaja-type theorem for the bivariate case were obtained by the weighted A-statistical convergence. A Korovkin-type theorem is provided for the bivariate case, and the convergence rate of blending bivariate operators to a function $r(u,v)$ was computed by the modulus of continuity. Moreover, the convergence behavior of blending univariate and bivariate $(\alpha ,\lambda ,s)$-Bernstein operators and the behavior of blending univariate and bivariate polynomials are demonstrated by computer graphics.

2. Preliminaries

In this section, we give the definitions of $\alpha$-Bernstein, $\lambda$-Bernstein, and blending $(\alpha ,\lambda ,s)$-Bernstein basis functions. Furthermore, the definitions of $\alpha$-Bernstein, $\lambda$-Bernstein, and blending $(\alpha ,\lambda ,s)$-Bernstein operators and all needed results are provided.

Let throughout the paper the binomial coefficients be given by the formula:

$$\left(\genfrac{}{}{0pt}{}{p}{i}\right)=\left\{\begin{array}{cc}\frac{p!}{i!(p-i)!},\hfill & 0\le i\le p,\hfill \\ \overline{0},\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

The original $\alpha$-Bernstein operators (see [5]) are defined as:

$$T_{p,\alpha }(r;u)=\sum _{i=0}^p{w}_{p,i}^{\left(\alpha \right)}\left(u\right)r\left(\frac{i}{p}\right),$$

where $w_{1,0}^{\left(\alpha \right)}\left(u\right)=1-u$, $w_{1,1}^{\left(\alpha \right)}\left(u\right)=u$, and the $\alpha$-Bernstein basis is given as:

$$\begin{array}{ccc}\hfill w_{p,i}^{\left(\alpha \right)}\left(u\right)& =& \left[(1-\alpha )\left(\genfrac{}{}{0pt}{}{p-2}{i}\right)u+(1-\alpha )\left(\genfrac{}{}{0pt}{}{p-2}{i-2}\right)(1-u)\right.\hfill \\ & & \left.+\alpha \left(\genfrac{}{}{0pt}{}{p}{i}\right)u(1-u)\right]u^{i-1}{(1-u)}^{p-i-1},\hfill \end{array}$$

for $\alpha ,u\in [0,1],$ $p\ge 2$, $r\left(u\right)\in C[0,1]$.

Furthermore, the $\lambda$-Bernstein operators are given as (see [10]):

$${\mathcal{B}}_{p,\lambda }(r;u)=\sum _{i=0}^p{\tilde{b}}_{p,i}\left(u\right)r\left(\frac{i}{p}\right),$$

(1)

where the $\lambda$-Bernstein basis is given as:

$${\tilde{b}}_{p,i}(\lambda ;u)=\left\{\begin{array}{cc}{b}_{p,0}\left(u\right)-\frac{\lambda }{p+1}{b}_{p+1,1}\left(u\right),\hfill & \mathrm{if}i=0,\hfill \\ \\ b_{p,i}\left(u\right)+\lambda \left(\frac{p-2i+1}{p^2-1}{b}_{p+1,i}\left(u\right)\right)\hfill \\ \\ -\lambda \left(\frac{p-2k-1}{p^2-1}{b}_{p+1,i+1}\left(u\right)\right),\hfill & \mathrm{if}1\le i\le p-1,\hfill \\ \\ b_{p,p}\left(u\right)-\frac{\lambda }{p+1}{b}_{p+1,p}\left(u\right),\hfill & \mathrm{if}i=p.\hfill \end{array}\right.$$

(2)

On the other hand, in [11], the authors introduced generalized blending-type $\alpha$-Bernstein operators by implementing a positive integer s as:

$$\begin{array}{ccc}\hfill {\mathcal{L}}_p^{\alpha ,s}(r;u)& =& \sum _{i=0}^p\left\{(1-\alpha )\left(\genfrac{}{}{0pt}{}{p-s}{i-s}\right)u^{i-s+1}{(1-u)}^{p-i}\right.\hfill \\ & & +(1-\alpha )\left(\genfrac{}{}{0pt}{}{p-s}{i}\right)u^i{(1-u)}^{p-s-i+1}\hfill \\ & & \left.+\alpha \left(\genfrac{}{}{0pt}{}{p}{i}\right)u^i{(1-u)}^{p-i}\right\}r\left(\frac{i}{p}\right),\mathrm{for}p\ge s\hfill \end{array}$$

and:

$${\mathcal{L}}_p^{\alpha ,s}(r;u)=\sum _{i=0}^p\left(\genfrac{}{}{0pt}{}{p}{i}\right)u^i{(1-u)}^{p-i}r\left(\frac{i}{p}\right),\mathrm{for}p<s$$

which depend on shape parameter $\alpha ,$ where $u,\alpha \in [0,1]$, $r\left(u\right)\in C[0,1]$.

Finally, blending-type $(\alpha ,\lambda ,s)$-Bernstein operators were constructed in [32] as follows:

$${\mathcal{L}}_{p,\lambda }^{(\alpha ,s)}(r;u)=\sum _{i=0}^p{\tilde{b}}_{p,i}^{\alpha ,s}(\lambda ;u)r\left(\frac{i}{p}\right),$$

(3)

where $0\le \alpha \le 1$, $-1\le \lambda \le 1$ and s is a positive integer, and the blending-type $(\alpha ,\lambda ,s)$ basis is given as:

$${\tilde{b}}_{p,i}^{\alpha ,s}(\lambda ;u)=\left\{\begin{array}{cc}{\tilde{b}}_{p,i}(\lambda ;u),\hfill & \mathrm{if}p<s\hfill \\ (1-\alpha )\left[x{\tilde{b}}_{p-s,i-s}(\lambda ;u)+(1-u){\tilde{b}}_{p-s,i}(\lambda ;u)\right]\hfill \\ +\alpha {\tilde{b}}_{p,i}(\lambda ;u),\hfill & \mathrm{if}p\ge s\hfill \end{array}\right.$$

and ${\tilde{b}}_{p,i}(\lambda ;u)$ defined in Equation (2).

In [32], the authors proposed an alternative representation for (3) as:

$${\mathcal{L}}_{p,\lambda }^{(\alpha ,s)}(r;u)=\left\{\begin{array}{cc}{\mathcal{B}}_{p,\lambda }(r;u),\hfill & \mathrm{if}p<s\hfill \\ \\ {\mathcal{B}}_{p,\lambda }^{\alpha ,s}(r;u),\hfill & \mathrm{if}p\ge s.\hfill \end{array}\right.$$

(4)

Here, ${\mathcal{B}}_{p,\lambda }(r;u)$ is given by Equation (1) and ${\mathcal{B}}_{p,\lambda }^{\alpha ,s}(r;u)$ is defined by:

$${\mathcal{B}}_{p,\lambda }^{\alpha ,s}(r;u)=(1-\alpha ){\mathcal{B}}_{p,\lambda }^{s,(★)}(r;u)+\alpha {\mathcal{B}}_{p,\lambda }(r;u),$$

where:

$${\mathcal{B}}_{p,\lambda }^{s,(★)}(r;u)=\left[u\sum _{i=0}^p{\tilde{b}}_{p-s,i-s}(\lambda ;u)+(1-u)\sum _{i=0}^p{\tilde{b}}_{p-s,i}(\lambda ;u)\right]r\left(\frac{i}{p}\right).$$

In [32], the representation (4) was also explicitly written as:

$$\begin{array}{ccc}\hfill {\mathcal{B}}_{p,\lambda }(r;u)& =& \sum _{i=0}^p{b}_{p,i}\left(u\right)r\left(\frac{i}{p}\right)\hfill \\ & & +\lambda \sum _{i=0}^{p-1}\frac{p-2k-1}{p^2-1}{b}_{p+1,i+1}\left(u\right)\left[r\left(\frac{i+1}{p}\right)-r\left(\frac{i}{p}\right)\right]\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {\mathcal{B}}_{p,\lambda }^{s,(★)}(r;u)& =& \sum _{i=0}^{p-s}{b}_{p-s,i}\left(u\right)\left[ur\left(\frac{i+s}{p}\right)+(1-u)r\left(\frac{i}{p}\right)\right]\hfill \\ & & +\lambda u\sum _{i=0}^{p-s-1}\frac{p-s-2k-1}{{(p-s)}^2-1}{b}_{p-s+1,i+1}\left(u\right)\hfill \\ & & \times \left[r\left(\frac{i+s+1}{p}\right)-r\left(\frac{i+s}{p}\right)\right]\hfill \\ & & +\lambda (1-u)\sum _{i=0}^{p-s-1}\frac{p-s-2k-1}{{(p-s)}^2-1}{b}_{p-s+1,i+1}\left(u\right)\hfill \\ & & \times \left[r\left(\frac{i+1}{p}\right)-r\left(\frac{i}{p}\right)\right].\hfill \end{array}$$

Lemma 1

([32]Theorem2).If $p\ge s$, for any $0\le \alpha \le 1$ and $-1\le \lambda \le 1$, we have the following identities:

$$\begin{array}{ccc}\hfill \left(i\right){\mathcal{L}}_{p,\lambda }^{(\alpha ,s)}(1;u)& =& 1;\hfill \\ \hfill \left(ii\right){\mathcal{L}}_{p,\lambda }^{(\alpha ,s)}(t;u)& =& u+(1-\alpha )\lambda \left[\frac{1-2u+u^{p-s+1}-{(1-u)}^{p-s+1}}{p(p-s-1)}\right]\hfill \\ & & +\alpha \lambda \left[\frac{1-2u+u^{p+1}-{(1-u)}^{p+1}}{p(p-1)}\right];\hfill \\ \hfill \left(iii\right){\mathcal{L}}_{p,\lambda }^{(\alpha ,s))}(t^2;u)& =& u^2+\frac{\left[p+(1-\alpha )s(s-1)\right]u(1-u)}{p^2}\hfill \\ & & +\frac{\alpha \lambda }{p}\left[\frac{2u-4u^2+2u^{p+1}}{(p-1)}\right]\hfill \\ & & +\frac{(1-\alpha )\lambda }{p}\left[\frac{2u-4u^2+2u^{p-s+1}}{(p-s-1)}\right]\hfill \\ & & +\frac{\alpha \lambda }{p^2}\left[\frac{u^{p+1}+{(1-u)}^{p+1}-1}{(p-1)}\right]\hfill \\ & & +\frac{(1-\alpha )\lambda }{p^2}\left[\frac{u^{p-s+1}+{(1-u)}^{p-s+1}-1}{(p-s-1)}\right]\hfill \\ & & +\left[\frac{2su(u^{p-s+1}-{(1-u)}^{p-s+1})}{(p-s-1)}\right].\hfill \end{array}$$

3. Statistical Convergence of Univariate Blending $(\mathbf{\alpha },\mathbf{\lambda },\mathbf{s})$-Bernstein Operators

In this section, certain statistical convergence results for blending ${\mathcal{L}}_{p,\lambda }^{(\alpha ,s)}$ operators are provided. First, the needed standard notions and notations that were also given in the papers [33,34] are provided.

Definition 1.

Let vertical bars denote the cardinality of the enclosed set, then the natural density of $K_p$ is represented by $\delta \left(K\right)={lim}_p\frac{1}{p}\left|K_p\right|$ provided that the limit exists, where $K_p=\{i\le p:i\in K\}$, $K\subseteq {\mathbb{N}}_0:=\mathbb{N}\cup \left\{0\right\}.$

Definition 2.

A sequence $u=\left(u_p\right)$ of numbers is called statistically convergent to a number E, denoted by st-lim_p $u=E$, if, for each $ϵ>0$,

$$\begin{array}{c}\hfill \delta \{p:p\in \mathbb{N}\mathit{and}|u_p-E|\geqq ϵ\}=0.\end{array}$$

The A-transform of u denoted by $Au:=\left\{{\left(Au\right)}_p\right\}$ is defined as ${\left(Au\right)}_p={\sum }_{i=0}^{\infty }{a}_{pi}{u}_i$ for a given non-negative infinite summability matrix $A=\left(a_{pi}\right),p,i\in \mathbb{N}$. If ${lim}_p{\left(Au\right)}_p=E$ whenever ${lim}_p{u}_p=E$, we say that A is a regular method. Then, sequence $u=\left(u_p\right)$ is said to be A-statistically convergent to E, denoted by st_A-lim $u=E$, provided that for each $ϵ>0$,

$$\begin{array}{c}\hfill \sum _{i:|u_p-E|\geqq ϵ}{a}_{pi}=0(p\to \infty ).\end{array}$$

Definition 3.

Let $Q_p={\sum }_{i=0}^p{q}_k\to \infty$ as $p\to \infty ,$ where $q>0$. Then, $u=\left(u_p\right)$ is called weighted A-statistically convergent to E, if, for every $\epsilon >0$,

$$\begin{array}{c}\hfill \underset{p\to \infty }{lim}\frac{1}{Q_p}\sum _{i=0}^p{q}_k\sum _{j:|u-E|\geqq ϵ}{a}_{ij}=0.\end{array}$$

This relation is denoted by ${ws}_A-limu=E$ in this case.

The following Korovkin type theorem is given by [35]:

Theorem 1.

Let $\alpha \in [0,1],\lambda \in [-1,1],$ $A=\left(a_{pk}\right)$ be a non-negative regular weighted summability matrix for $p,i\in \mathbb{N}$ and $q>0$ such that $Q_p={\sum }_{i=0}^p{q}_k\to \infty$ as $p\to \infty$. Then, the following relation is obtained for each $r\in C[0,1]$:

$$\begin{array}{c}\hfill {ws}_A-\underset{p\to \infty }{lim}{\parallel {\mathcal{L}}_{p,\lambda }^{(\alpha ,s)}(r;u)-r\left(u\right)\parallel }_{C[0,1]}=0.\end{array}$$

Proof. 

Let the hypothesis of the theorem be satisfied and the sequence of functions $e_j\left(u\right)=u^j$ be given, where $j\in \{0,1,2\}$ and $u\in [0,1]$. It is sufficient to satisfy the following equality for $j=0,1,2$:

$$\begin{array}{c}\hfill {ws}_A-\underset{p\to \infty }{lim}{\parallel {\mathcal{L}}_{p,\lambda }^{(\alpha ,s)}(e_j;u)-e_j\parallel }_{C[0,1]}=0.\end{array}$$

It is clear from Lemma 1 that:

$$\begin{array}{c}\hfill {ws}_A-\underset{p\to \infty }{lim}{\parallel {\mathcal{L}}_{p,\lambda }^{(\alpha ,s)}(e_0;u)-e_0\parallel }_{C[0,1]}=0\end{array}$$

is satisfied. Furthermore, the following relation is satisfied:

$$\begin{array}{cc}\hfill \parallel {\mathcal{L}}_{p,\lambda }^{(\alpha ,s)}(e_1;u)-e_1{\parallel }_{C[0,1]}& =\underset{u\in [0,1]}{sup}|(1-\alpha )\lambda \left[\frac{1-2u+u^{p-s+1}-{(1-u)}^{p-s+1}}{p(p-s-1)}\right]\hfill \\ \hfill & +\alpha \lambda \left[\frac{1-2u+u^{p+1}-{(1-u)}^{p+1}}{p(p-1)}\right]|\le {\mathrm{\Psi }}_{p,s,1}(u;\alpha )\hfill \end{array}$$

by [32], Corollary 2, where:

$${\mathrm{\Psi }}_{p,s,1}(u;\alpha )=\left\{\begin{array}{cc}\frac{1-2u+u^{p+1}-{(1-u)}^{p+1}}{p(p-1)},\hfill & \mathrm{if}p<s\hfill \\ \\ (1-\alpha )\left[\frac{1+2u+u^{p-s+1}+{(1-u)}^{p-s+1}}{p(p-s-1)}\right]\hfill \\ +\alpha \left[\frac{1+2u+u^{p+1}+{(1-u)}^{p+1}}{p(p-1)}\right],\hfill & \mathrm{if}p\ge s.\hfill \end{array}\right.$$

(5)

By defining the following sets:

$$\begin{array}{ccc}& & {\mathcal{H}}_1:=\left\{p\in \mathbb{N}:\parallel {\mathcal{L}}_{p,\lambda }^{(\alpha ,s)}(e_1;u)-e_1{\parallel }_{C[0,1]}\geqq \overline{ϵ}\right\},\hfill \\ & & {\mathcal{H}}_2:=\left\{p\in \mathbb{N}:{\mathrm{\Psi }}_{p,s,1}(u;\alpha )\geqq ϵ-\overline{ϵ}\right\}\hfill \end{array}$$

and choosing a number $ϵ>0$ for a given $\overline{ϵ}>0$ such that $ϵ<\overline{ϵ}$, then we see that the inclusion ${\mathcal{H}}_1\subset {\mathcal{H}}_2$ is satisfied and:

$$\begin{array}{c}\hfill \frac{1}{Q_p}\sum _{i=0}^p{q}_k\sum _{l\in {\mathcal{H}}_1}{a}_{il}\le \frac{1}{Q_p}\sum _{i=0}^p{q}_k\sum _{l\in {\mathcal{H}}_2}{a}_{il}\end{array}$$

(6)

for all $p\in \mathbb{N}$. Passing the limit as $p\to \infty$ in (6), we have:

$$\begin{array}{c}\hfill {ws}_A-\underset{p\to \infty }{lim}{\parallel {\mathcal{L}}_{p,\lambda }^{(\alpha ,s)}(e_1;u)-e_1\parallel }_{C[0,1]}=0.\end{array}$$

Using a similar manner, we obtain:

$$\begin{array}{c}\hfill {ws}_A-\underset{p\to \infty }{lim}{\parallel {\mathcal{L}}_{p,\lambda }^{(\alpha ,s)}(e_2;u)-e_2\parallel }_{C[0,1]}=0.\end{array}$$

□

Definition 4.

Let $q>0$ and $A=\left(a_{pk}\right)$ be a weighted non-negative regular summability matrix such that $q_0>0$ and $Q_p={\sum }_{i=0}^p{q}_k\to \infty$ as $p\to \infty$. Let $\left(u_p\right)$ be a positive non-decreasing sequence; a sequence $u=\left(u_p\right)$ is weighted A-statistically convergent to E with the rate $o\left(u_p\right)$ if:

$$\begin{array}{c}\hfill \underset{p\to \infty }{lim}\frac{1}{u_p{Q}_p}\sum _{i=0}^p{q}_k\sum _{j:|x_l-E|\geqq ϵ}{a}_{ij}=0.\end{array}$$

This fact is expressed as:

$$\begin{array}{c}\hfill [sta{t}_A,q_p]-o\left(u_p\right)=u_p-E.\end{array}$$

We give a rate of convergence for ${\mathcal{L}}_{p,\lambda }^{(\alpha ,s)}$ by a non-negative regular weighted summability matrix in the following theorem.

Theorem 2.

Let $A=\left(a_{pk}\right)$ be a non-negative regular weighted summability matrix. If

$$\begin{array}{c}\hfill w(r,{\phi }_p)=[sta{t}_A,q_p]-o\left(u_p\right)on[0,1],where{\phi }_p=\sqrt{\parallel {\mathcal{L}}_{p,\lambda }^{(\alpha ,s)}({(t-u)}^2;u){\parallel }_{C[0,1]}}\end{array}$$

is satisfied, then:

$$\begin{array}{c}\hfill \parallel {\mathcal{L}}_{p,\lambda }^{(\alpha ,s)}{(r;u)-r\left(u\right)\parallel }_{C[0,1]}=[sta{t}_A,q_p]-o\left(u_p\right)\end{array}$$

is satisfied for all bounded $r\in C[0,1].$

Proof. 

Let $r\left(u\right)\in C[0,1]$, then:

$$\begin{array}{cc}\hfill |{\mathcal{L}}_{p,\lambda }^{(\alpha ,s)}(r;u)-r\left(u\right)|& \le |{\mathcal{L}}_{p,\lambda }^{(\alpha ,s)}\left(\left|r\right(t)-r(u\left)\right|;u\right)+\psi |{\mathcal{L}}_{p,\lambda }^{(\alpha ,s)}(1;u)-1|\hfill \\ \hfill & \le \omega (r,\delta ){\mathcal{L}}_{p,\lambda }^{(\alpha ,s)}\left(\frac{|t-u|}{\delta }+1;u\right)\hfill \\ \hfill & =\omega (r,\delta ){\mathcal{L}}_{p,\lambda }^{(\alpha ,s)}(1;u)+\omega (r,\delta )\frac{1}{{\delta }^2}{\mathcal{L}}_{p,\lambda }^{(\alpha ,s)}\left({(t-u)}^2;u\right)\hfill \end{array}$$

for any $u,s\in [0,1]$, where $\psi ={sup}_{u\in [0,1]}\left|r\left(u\right)\right|$. Let $\delta :={\phi }_p$ for all $p\in \mathbb{N}$. Taking the supremum over $u\in [0,\infty )$ on both sides, we obtain:

$$\begin{array}{c}\hfill \parallel {\mathcal{L}}_{p,\lambda }^{(\alpha ,s)}{(r;u)-r\left(u\right)\parallel }_{C[0,1]}\le \omega (r,{\phi }_p)+\omega (r,{\phi }_p)\frac{1}{{\phi }_p^2}{\parallel {\mathcal{L}}_{p,\lambda }^{(\alpha ,s)}({(t-u)}^2;u)\parallel }_{C[0,1]}=2\omega (r,{\phi }_p).\end{array}$$

Consider following sets for a given $ϵ>0$:

$$\begin{array}{ccc}\hfill {\mathcal{T}}_1& =& \left\{p:\parallel {\mathcal{L}}_{p,\lambda }^{(\alpha ,s)}{(r;u)-r\left(u\right)\parallel }_{C[0,1]}\ge ϵ\right\},\hfill \\ \hfill {\mathcal{T}}_2& =& \left\{p:\omega (r,{\phi }_p)\ge \frac{ϵ}{2}\right\}.\hfill \end{array}$$

The following inequality is clearly satisfied:

$$\begin{array}{c}\hfill \frac{1}{u_p{Q}_p}\sum _{i=0}^p\sum _{l\in {\mathcal{T}}_1}{q}_k{a}_{il}\le \frac{1}{u_p{Q}_p}\sum _{i=0}^p\sum _{l\in {\mathcal{T}}_2}{q}_k{a}_{il}.\end{array}$$

We are led to the following fact by the hypothesis that:

$$\begin{array}{c}\hfill \parallel {\mathcal{L}}_{p,\lambda }^{(\alpha ,s)}{(r;u)-r\left(u\right)\parallel }_{C[0,1]}=[sta{t}_A,q_p]-o\left(u_p\right)\end{array}$$

as asserted by Theorem 2. □

The following Voronovskaja-type theorem is obtained by the weighted A-statistical convergence.

Theorem 3.

Let $A=\left(a_{pk}\right)$ be a non-negative regular weighted summability matrix, $\left(u_p\right)$ be a sequence of real numbers such that ${ws}_A-lim{u}_p=0$ and:

$$\begin{array}{c}\hfill {\mathcal{L}}_{p,\lambda }^{(\alpha ,s,\ast )}(r;u)=(1+u_p){\mathcal{L}}_{p,\lambda }^{(\alpha ,s)}(r;u),\end{array}$$

where ${\mathcal{L}}_{p,\lambda }^{(\alpha ,s,\ast )}:C_B[0,1]\to C[0,1].$ Then:

$$\begin{array}{c}\hfill {ws}_A-\underset{p\to \infty }{lim}p\left\{{\mathcal{L}}_{p,\lambda }^{(\alpha ,s,\ast )}(r;u)-r\left(u\right)\right\}=(1-\alpha )\lambda (\frac{1}{2}-u)r^{\prime }\left(u\right)+\frac{u-u^2}{2}{r}^{″}\left(u\right)\end{array}$$

for all $r\left(u\right)\in C_B[0,1]$ and $r^{\prime },r^{″}\in C_B[0,1]$.

Proof. 

Let $u\in [0,1]$ and $r^{\prime },r^{″}\in C_B[0,1]$. Applying ${\mathcal{L}}_{p,\lambda }^{(\alpha ,s,\ast )}$ to both sides of Taylor’s expansion,

$$\begin{array}{cc}\hfill {\mathcal{L}}_{p,\lambda }^{(\alpha ,s,\ast )}(r;u)-r\left(u\right)& =r^{{}^{\prime }}\left(u\right){\mathcal{L}}_{p,\lambda }^{(\alpha ,s,\ast )}(t-u;u)+\frac{r^{{}^{″}}\left(u\right)}{2}{\mathcal{L}}_{p,\lambda }^{(\alpha ,s,\ast )}({(t-u)}^2;u)\hfill \\ & +{\mathcal{L}}_{p,\lambda }^{(\alpha ,s,\ast )}({(t-u)}^2ϵ(u,t);u),\hfill \end{array}$$

which yields:

$$\begin{array}{cc}\hfill p\left\{{\mathcal{L}}_{p,\lambda }^{(\alpha ,s,\ast )}(r;u)-r\left(u\right)\right\}& =p{r}^{{}^{\prime }}\left(u\right)(1+u_p){\mathcal{L}}_{p,\lambda }^{(\alpha ,s)}(t-u;u)+\frac{p}{2}{r}^{{}^{″}}\left(u\right)(1+u_p){\mathcal{L}}_{p,\lambda }^{(\alpha ,s)}({(t-u)}^2;u)\hfill \\ \hfill & +p(1+u_p){\mathcal{L}}_{p,\lambda }^{(\alpha ,s)}({(t-u)}^2ϵ(u,t);u).\hfill \end{array}$$

We also have the following relations:

$$\begin{array}{cc}\hfill & \left|p\left\{{\mathcal{L}}_{p,\lambda }^{(\alpha ,s,\ast )}(r;u)-r\left(u\right)\right\}-r^{{}^{\prime }}\left(u\right)\left(p(1-\alpha )\lambda \left[\frac{1-2u+u^{p-s+1}-{(1-u)}^{p-s+1}}{p-s-1}\right]\right.\right.\hfill \\ \hfill & \left.+\alpha \lambda \left[\frac{1-2u+u^{p+1}-{(1-u)}^{p+1}}{p-1}\right]\right)-r^{{}^{″}}\left(u\right)\left(\frac{\left[p+(1-\alpha )s(s-1)\right]u(1-u)}{2p}\right.\hfill \\ \hfill & +\alpha \lambda \left[\frac{u^{p+1}+{(1-u)}^{p+1}-1}{p(p-1)}\right]+(1-\alpha )\lambda \left[\frac{u^{p-s+1}+{(1-u)}^{p-s+1}-1}{p(p-s-1)}\right]\hfill \\ \hfill & \left.\left.+2(1-\alpha )\lambda su\left[\frac{u^{p-s+1}-{(1-u)}^{p-s+1}}{p(p-s-1)}\right]+\alpha \lambda \frac{{(1-u)}^{p+1}}{p-1}+(1-\alpha )\lambda \frac{{(1-u)}^{p-s+1}}{p-s-1}\right)\right|\hfill \\ \hfill & =p{r}^{{}^{\prime }}\left(u\right)u_p{\mathcal{L}}_{p,\lambda }^{(\alpha ,s)}(t-u;u)+\frac{p}{2}{r}^{{}^{″}}\left(u\right)u_p{\mathcal{L}}_{p,\lambda }^{(\alpha ,s)}({(t-u)}^2;u)+p(1+u_p){\mathcal{L}}_{p,\lambda }^{(\alpha ,s)}({(t-u)}^2ϵ(u,t);u)\hfill \\ \hfill & \le u_p\left\{n{r}^{{}^{\prime }}\left(u\right){\mathrm{\Psi }}_{p,s,1}+p\frac{r^{{}^{″}}\left(u\right)}{2}{\mathrm{\Psi }}_{p,s,2}\right\}+p(1+u_p){\mathcal{L}}_{p,\lambda }^{(\alpha ,s)}({(t-u)}^2ϵ(u,t);u)\hfill \\ \hfill & \le u_p\left\{p{D}_1{\mathrm{\Psi }}_{p,s,1}+p{D}_2{\mathrm{\Psi }}_{p,s,2}\right\}+p(1+u_p){\mathcal{L}}_{p,\lambda }^{(\alpha ,s)}({(t-u)}^2ϵ(u,t);u),\hfill \end{array}$$

where $D_1={sup}_{u\in [0,1]}|r^{\prime }\left(u\right)|,D_2={sup}_{u\in [0,1]}\left|r^{″}\left(u\right)\right|,$${\mathrm{\Psi }}_{p,s,1}(u;\alpha )$ is defined in (5) and:

$${\mathrm{\Psi }}_{p,s,2}(u;\alpha )=\left\{\begin{array}{c}\frac{u(1-u)}{p}+\frac{2{(1-u)}^{p+1}}{p(p-1)}\hfill \\ +\frac{u^{p+1}+{(1-u)}^{p+1}+1}{p^2(p-1)},\hfill & \mathrm{if}p<s\hfill \\ \\ \frac{\left[p+(1-\alpha )s(s-1)\right]u(1-u)}{p^2}+\frac{2\alpha {(1-u)}^{p+1}}{p(p-1)}\hfill \\ +\frac{2(1-\alpha ){(1-u)}^{p-s+1}}{p(p-s-1)}+\alpha \left[\frac{u^{p+1}+{(1-u)}^{p+1}+1}{p^2(p-1)}\right]\hfill \\ +(1-\alpha )\left[\frac{u^{p-s+1}+{(1-u)}^{p-s+1}+1}{p^2(p-s-1)}\right]\hfill \\ +(1-\alpha )2sx\left[\frac{u^{p-s+1}+{(1-u)}^{p-s+1}}{p^2(p-s-1)}\right],\hfill & \mathrm{if}p\ge s.\hfill \end{array}\right.$$

Moreover, by Theorem 1, we obtain:

$$\begin{array}{c}\hfill {ws}_A-\underset{p\to \infty }{lim}p\left({\mathcal{L}}_{p,\lambda }^{(\alpha ,s)}({(t-u)}^2ϵ(u,t);u)\right)=0.\end{array}$$

We obtain the desired result since ${ws}_A-lim{u}_p=0$. □

4. Bivariate Operators

In this part, we construct blending bivariate $(\alpha ,\lambda ,s)$-Bernstein operators and obtain certain approximation theorems. Let $\mathcal{I}=[0,1]\times [0,1]i$$(u,v)\in \mathcal{I}$ and $C\left(\mathcal{I}\right)$ be the set of all real-valued continuous functions on $\mathcal{I}$ with the norm:

$$\begin{array}{c}\hfill {\Vert r\Vert }_{C\left(\mathcal{I}\right)}=\underset{(u,v)\in \mathcal{I}}{sup}\left|r(u,v)\right|.\end{array}$$

The blending bivariate $(\alpha ,\lambda ,s)$-Bernstein operators are defined as:

$${\mathcal{L}}_{p,q,{\lambda }_1,{\lambda }_2}^{({\alpha }_1,{\alpha }_2,s)}(r;u,v)=\left\{\begin{array}{cc}{\mathcal{B}}_{p,q}^{{\lambda }_1,{\lambda }_2}(r;u,v)\hfill & \mathrm{if}p,q<s\hfill \\ \\ {\mathcal{B}}_{p,q,{\lambda }_1,{\lambda }_2}^{{\alpha }_1,{\alpha }_2,s}(r;u,v),\hfill & \mathrm{if}p,q\ge s.\hfill \end{array}\right.$$

(7)

Here, ${\mathcal{B}}_{p,q}^{{\lambda }_1,{\lambda }_2}(r;u,v)={\mathcal{B}}_{p,{\lambda }_1}(r;u){\mathcal{B}}_{q,{\lambda }_2}(r;v)$ and ${\mathcal{B}}_{p,q,{\lambda }_1,{\lambda }_2}^{{\alpha }_1,{\alpha }_2,s}(r;u,v)$ defined by:

$${\mathcal{B}}_{p,q,{\lambda }_1,{\lambda }_2}^{{\alpha }_1,{\alpha }_2,s}(r;u,v)=(1-{\alpha }_1)(1-{\alpha }_2){\mathcal{B}}_{p,{\lambda }_1}^{s,(★)}(r;u){\mathcal{B}}_{q,{\lambda }_2}^{s,(★)}(r;v)+{\alpha }_1{\alpha }_2{\mathcal{B}}_{p,{\lambda }_1}(r;u){\mathcal{B}}_{q,{\lambda }_2}(r;v),$$

where:

$${\mathcal{B}}_{p,{\lambda }_1}^{s,(★)}(r;u)=\left[u\sum _{i=0}^p{\tilde{b}}_{p-s,i-s}(\lambda ;u)+(1-u)\sum _{i=0}^p{\tilde{b}}_{p-s,i}({\lambda }_1;u)\right]r\left(\frac{i}{p}\right);$$

$${\mathcal{B}}_{q,{\lambda }_2}^{s,(★)}(r;v)=\left[v\sum _{j=0}^q{\tilde{b}}_{q-s,j-s}({\lambda }_2;v)+(1-v)\sum _{j=0}^q{\tilde{b}}_{q-s,j}({\lambda }_2;v)\right]r\left(\frac{j}{q}\right).$$

Lemma 2.

Let $p,q\ge s$, for any $0\le {\alpha }_1,{\alpha }_2\le 1$ and $-1\le {\lambda }_1,{\lambda }_2\le 1,$ then:

$$\begin{array}{cc}\hfill \left(i\right){\mathcal{L}}_{p,q,{\lambda }_1,{\lambda }_2}^{({\alpha }_1,{\alpha }_2,s)}(e_{00};u,v)& =1;\hfill \\ \hfill \left(ii\right){\mathcal{L}}_{p,q,{\lambda }_1,{\lambda }_2}^{({\alpha }_1,{\alpha }_2,s)}(e_{10};u,v)& =u+(1-{\alpha }_1){\lambda }_1\left[\frac{1-2u+u^{p-s+1}-{(1-u)}^{p-s+1}}{p(p-s-1)}\right]\hfill \\ \hfill & +{\alpha }_1{\lambda }_1\left[\frac{1-2u+u^{p+1}-{(1-u)}^{p+1}}{p(p-1)}\right];\hfill \\ \hfill \left(iii\right){\mathcal{L}}_{p,q,{\lambda }_1,{\lambda }_2}^{({\alpha }_1,{\alpha }_2,s)}(e_{01};u,v)& =v+(1-{\alpha }_2){\lambda }_2\left[\frac{1-2v+v^{q-s+1}-{(1-v)}^{q-s+1}}{q(q-s-1)}\right]\hfill \\ \hfill & +{\alpha }_2{\lambda }_2\left[\frac{1-2v+v^{q+1}-{(1-v)}^{q+1}}{q(q-1)}\right];\hfill \\ \hfill \left(iv\right){\mathcal{L}}_{p,q,{\lambda }_1,{\lambda }_2}^{({\alpha }_1,{\alpha }_2,s)}(e_{20};u,v)& =u^2+\frac{\left[p+(1-{\alpha }_1)s(s-1)\right]u(1-u)}{p^2}+\frac{{\alpha }_1{\lambda }_1}{p}\left[\frac{2u-4u^2+2u^{p+1}}{(p-1)}\right]\hfill \\ & +\frac{(1-{\alpha }_1){\lambda }_1}{p}\left[\frac{2u-4u^2+2u^{p-s+1}}{(p-s-1)}\right]+\frac{{\alpha }_1{\lambda }_1}{p^2}\left[\frac{u^{p+1}+{(1-u)}^{p+1}-1}{(p-1)}\right]\hfill \\ \hfill & +\frac{(1-{\alpha }_1){\lambda }_1}{p^2}\left[\frac{u^{p-s+1}+{(1-u)}^{p-s+1}-1}{(p-s-1)}\right]\hfill \\ \hfill & +\frac{2su(u^{p-s+1}-{(1-u)}^{p-s+1})}{(p-s-1)};\hfill \\ \hfill \left(v\right){\mathcal{L}}_{p,q,{\lambda }_1,{\lambda }_2}^{({\alpha }_1,{\alpha }_2,s)}(e_{02};u,v)& =v^2+\frac{\left[q+(1-{\alpha }_2)s(s-1)\right]v(1-v)}{q^2}+\frac{{\alpha }_2{\lambda }_2}{q}\left[\frac{2v-4v^2+2v^{q+1}}{(q-1)}\right]\hfill \\ \hfill & +\frac{(1-{\alpha }_2){\lambda }_2}{q}\left[\frac{2v-4v^2+2v^{q-s+1}}{(q-s-1)}\right]+\frac{{\alpha }_2{\lambda }_2}{q^2}\left[\frac{v^{q+1}+{(1-v)}^{q+1}-1}{(q-1)}\right]\hfill \\ \hfill & +\frac{(1-{\alpha }_2){\lambda }_2}{q^2}\left[\frac{v^{q-s+1}+{(1-v)}^{q-s+1}-1}{(q-s-1)}\right]\hfill \\ \hfill & +\frac{2sv(u^{q-s+1}-{(1-v)}^{q-s+1})}{(q-s-1)}.\hfill \end{array}$$

Theorem 4.

The sequence ${\mathcal{L}}_{p,q,{\lambda }_1,{\lambda }_2}^{({\alpha }_1,{\alpha }_2,s)}$ of operators converges uniformly to $r(u,v)$ by the weighted A-statistical convergence on $\mathcal{I}$ for each $r(u,v)\in C\left(\mathcal{I}\right)$.

Proof. 

The following relation must be verified for $(i,j)\in \left\{(0,0),(1,0),(0,1)\right\}:$

$$\begin{array}{c}\hfill {ws}_A-\underset{p,q\to \infty }{lim}{\mathcal{L}}_{p,q,{\lambda }_1,{\lambda }_2}^{({\alpha }_1,{\alpha }_2,s)}\left(e_{ij}(u,v);u,v\right)=u^i{v}^j\mathrm{converges}\mathrm{uniformly}\mathrm{on}\mathcal{I}.\end{array}$$

It can be seen that:

$$\begin{array}{c}\hfill {ws}_A-\underset{p,q\to \infty }{lim}{\mathcal{L}}_{p,q,{\lambda }_1,{\lambda }_2}^{({\alpha }_1,{\alpha }_2,s)}\left(e_{00}(u,v);u,v\right)=e_{00}.\end{array}$$

Furthermore:

$$\begin{array}{cc}\hfill & {ws}_A-\underset{p,q\to \infty }{lim}{\mathcal{L}}_{p,q,{\lambda }_1,{\lambda }_2}^{({\alpha }_1,{\alpha }_2,s)}\left(e_{10}(u,v);u,v\right)\hfill \\ \hfill & ={ws}_A-\underset{p\to \infty }{lim}\left\{(1-{\alpha }_1){\lambda }_1\left(\frac{1-2u+u^{p-s+1}-{(1-u)}^{p-s+1}}{p(p-s-1)}\right)\right.\hfill \\ & +u+\left.{\alpha }_1{\lambda }_1\left(\frac{1-2u+u^{p+1}-{(1-u)}^{p+1}}{p(p-1)}\right)\right\}=e_{10}(u,v);\hfill \\ \hfill & {ws}_A-\underset{p,q\to \infty }{lim}{\mathcal{L}}_{p,q,{\lambda }_1,{\lambda }_2}^{({\alpha }_1,{\alpha }_2,s)}\left(e_{01}(u,v);u,v\right)\hfill \\ \hfill & ={ws}_A-\underset{q\to \infty }{lim}\left((1-{\alpha }_2){\lambda }_2\left(\frac{1-2v+v^{q-s+1}-{(1-v)}^{q-s+1}}{q(q-s-1)}\right)\right.\hfill \\ \hfill & +\left.v+{\alpha }_2{\lambda }_2\left(\frac{1-2v+v^{q+1}-{(1-v)}^{q+1}}{q(q-1)}\right)\right\}=e_{01}(u,v)\hfill \end{array}$$

by Lemma 2, and:

$$\begin{array}{cc}\hfill & {ws}_A-\underset{p,q\to \infty }{lim}{\mathcal{L}}_{p,q,{\lambda }_1,{\lambda }_2}^{({\alpha }_1,{\alpha }_2,s)}\left(e_{02}(u,v)+e_{20}(u,v);u,v\right)\hfill \\ \hfill & ={ws}_A-\underset{p,q\to \infty }{lim}\left\{+\frac{{\alpha }_1{\lambda }_1}{p}\left[\frac{2u-4u^2+2u^{p+1}}{(p-1)}\right]+\frac{\left[p+(1-{\alpha }_1)s(s-1)\right]u(1-u)}{p^2}\right.\hfill \\ \hfill & +\frac{{\alpha }_2{\lambda }_2}{q}\left[\frac{2v-4v^2+2v^{q+1}}{(q-1)}\right]+\frac{(1-{\alpha }_1){\lambda }_1}{p}\left[\frac{2u-4u^2+2u^{p-s+1}}{(p-s-1)}\right]\hfill \\ \hfill & +\frac{(1-{\alpha }_1){\lambda }_1}{p^2}\left[\frac{u^{p-s+1}+{(1-u)}^{p-s+1}-1}{(p-s-1)}\right]+\frac{{\alpha }_2{\lambda }_2}{q^2}\left[\frac{v^{q+1}+{(1-v)}^{q+1}-1}{(q-1)}\right]\hfill \\ \hfill & +\frac{(1-{\alpha }_2){\lambda }_2}{q}\left[\frac{2v-4v^2+2v^{q-s+1}}{(q-s-1)}\right]+\frac{(1-{\alpha }_2){\lambda }_2}{q^2}\left[\frac{v^{q-s+1}+{(1-v)}^{q-s+1}-1}{(q-s-1)}\right]\hfill \\ \hfill & +\frac{{\alpha }_1{\lambda }_1}{p^2}\left[\frac{u^{p+1}+{(1-u)}^{p+1}-1}{(p-1)}\right]+\frac{\left[q+(1-{\alpha }_2)s(s-1)\right]v(1-v)}{q^2}\hfill \\ \hfill & \left.+\frac{2su(u^{p-s+1}-{(1-u)}^{p-s+1})}{(p-s-1)}+\frac{2sv(u^{q-s+1}-{(1-v)}^{q-s+1})}{(q-s-1)}+u^2+v^2\right\}\hfill \\ \hfill & =e_{02}(u,v)+e_{20}(u,v).\hfill \end{array}$$

Considering the above relations and Volkov’s result [36]:

$$\begin{array}{c}\hfill {ws}_A-\underset{p,q\to \infty }{lim}{\mathcal{L}}_{p,q,{\lambda }_1,{\lambda }_2}^{({\alpha }_1,{\alpha }_2,s)}\left(e_{ij}(u,v);u,v\right)=u^i{v}^j\end{array}$$

converges uniformly. □

The convergence rate of ${\mathcal{L}}_{p,q,{\lambda }_1,{\lambda }_2}^{({\alpha }_1,{\alpha }_2,s)}$ to $r(u,v)$ by the modulus of continuity is computed. The needed definitions are given below.

The complete modulus of continuity for a bivariate case is:

$$\omega (r,\delta )=sup\left\{|r(z,t)-r(u,v)|:\sqrt{{(z-u)}^2+{(t-v)}^2}\le \delta \right\}$$

for $f\in C\left({\mathcal{I}}_{ab}\right)$ and for every $(z,t),(u,v)\in {\mathcal{I}}_{ab}=[0,a]\times [0,b]$. Partial moduli of continuity with respect to u and v are:

$$\begin{array}{ccc}\hfill {\omega }_1(r,\delta )& =& sup\left\{|r(d_1,v)-r(d_2,v)|:v\in [0,b]\mathrm{and}|d_1-d_2|\le \delta \right\},\hfill \\ \hfill {\omega }_2(r,\delta )& =& sup\left\{|r(u,e_1)-r(u,e_2)|:u\in [0,a]\mathrm{and}|e_1-e_2|\le \delta \right\}.\hfill \end{array}$$

Peetre’s K-functional is given by:

$$\begin{array}{c}\hfill K(r,\delta )={\int }_{g\in C^2\left({\mathcal{I}}_{ab}\right)}\left\{{\parallel r-g\parallel }_{C\left({\mathcal{I}}_{ab}\right)}+\delta {\parallel g\parallel }_{C^2\left({\mathcal{I}}_{ab}\right)}\right\}\end{array}$$

for $\delta >0$, where $C^2\left({\mathcal{I}}_{ab}\right)$ is the space of functions of r such that r, $\frac{{\partial }^{j}f}{\partial u^j}$, and $\frac{{\partial }^{j}f}{\partial v^j}$$(j=1,2)$ in $C\left({\mathcal{I}}_{ab}\right)$.

Theorem 5.

Let $r(u,v)\in C\left(\mathcal{I}\right)$, then:

$$\left|{\mathcal{L}}_{p,q,{\lambda }_1,{\lambda }_2}^{({\alpha }_1,{\alpha }_2,s)}\left(r;u,v\right)-r\left(u,v\right)\right|\le 4\omega \left(r;\sqrt{{\mathrm{\Psi }}_{p,s,2}(u;{\alpha }_1)},\sqrt{{\mathrm{\Psi }}_{q,s,2}(v;{\alpha }_2)}\right)$$

for all $u\in \mathcal{I}$, where ${\mathrm{\Psi }}_{p,s,2}(u;{\alpha }_1)$ and ${\mathrm{\Psi }}_{q,s,2}(v;{\alpha }_2)$ are defined in Theorem 3.

Proof. 

The following inequalities are satisfied:

$$\begin{array}{cc}\hfill & |{\mathcal{L}}_{p,q,{\lambda }_1,{\lambda }_2}^{({\alpha }_1,{\alpha }_2,s)}\left(r;u,v\right)-r(u,v)|\hfill \\ \hfill & \le {\mathcal{L}}_{p,q,{\lambda }_1,{\lambda }_2}^{({\alpha }_1,{\alpha }_2,s)}\left(\left|r\right(z,t)-r(u,v\left)\right|;u,v\right)\hfill \\ \hfill & \le {\mathcal{L}}_{p,q,{\lambda }_1,{\lambda }_2}^{({\alpha }_1,{\alpha }_2,s)}\left(\omega \left(r;\sqrt{{(z-u)}^2+{(t-v)}^2}\right);u,v\right)\hfill \\ \hfill & \le \omega \left(r;\sqrt{{\mathrm{\Psi }}_{p,s,2}(u;{\alpha }_1)},\sqrt{{\mathrm{\Psi }}_{q,s,2}(v;{\alpha }_2)}\right)\frac{{\mathcal{L}}_{p,q,{\lambda }_1,{\lambda }_2}^{({\alpha }_1,{\alpha }_2,s)}\left(\sqrt{{(z-u)}^2+{(t-v)}^2};u,v\right)}{\sqrt{{\mathrm{\Psi }}_{p,s,2}(u;{\alpha }_1){\mathrm{\Psi }}_{q,s,2}(v;{\alpha }_2)}}\hfill \end{array}$$

because the defined bivariate $(\alpha ,\lambda ,s)$-Bernstein operators are linear and positive by the definition of operators and the complete modulus of continuity of $r(u,v)$. We also have:

$$\begin{array}{cc}\hfill & |{\mathcal{L}}_{p,q,{\lambda }_1,{\lambda }_2}^{({\alpha }_1,{\alpha }_2,s)}\left(r;u,v\right)-r(u,v)|\hfill \\ \hfill & \le \omega \left(r;\sqrt{{\mathrm{\Psi }}_{p,s,2}(u;{\alpha }_1)},\sqrt{{\mathrm{\Psi }}_{q,s,2}(v;{\alpha }_2)}\right)\hfill \\ \hfill & \times \left[1+\frac{1}{\sqrt{{\mathrm{\Psi }}_{p,s,2}(u;{\alpha }_1){\mathrm{\Psi }}_{q,s,2}(v;{\alpha }_2)}}{\left\{{\mathcal{L}}_{p,q,{\lambda }_1,{\lambda }_2}^{({\alpha }_1,{\alpha }_2,s)}\left({(z-u)}^2;u,v\right){\mathcal{L}}_{p,q,{\lambda }_1,{\lambda }_2}^{({\alpha }_1,{\alpha }_2,s)}\left({(t-v)}^2;u,v\right)\right\}}^{1/2}\right.\hfill \\ \hfill & +\left.\frac{\sqrt{{\mathcal{L}}_{p,q,{\lambda }_1,{\lambda }_2}^{({\alpha }_1,{\alpha }_2,s)}\left({(z-u)}^2;u,v\right)}}{\sqrt{{\mathrm{\Psi }}_{p,s,2}(u;{\alpha }_1)}}+\frac{\sqrt{{\mathcal{L}}_{p,q,{\lambda }_1,{\lambda }_2}^{({\alpha }_1,{\alpha }_2,s)}\left({(t-v)}^2;u,v\right)}}{\sqrt{{\mathrm{\Psi }}_{q,s,2}(v;{\alpha }_2)}}\right]\hfill \end{array}$$

by the Cauchy–Schwartz inequality. Choosing ${\mathrm{\Psi }}_{p,s,2}(u;{\alpha }_1)={\mathcal{L}}_{p,q,{\lambda }_1,{\lambda }_2}^{({\alpha }_1,{\alpha }_2,s)}({(z-u)}^2;u,v)$ and ${\mathrm{\Psi }}_{q,s,2}(v;{\alpha }_2)={\mathcal{L}}_{p,q,{\lambda }_1,{\lambda }_2}^{({\alpha }_1,{\alpha }_2,s)}({(t-v)}^2;u,v)$ for all $(u,v)\in \mathcal{I}$, we complete the proof. □

Theorem 6.

Let $r(u,v)\in C\left(I\right)$, then the following inequality holds:

$$\left|{\mathcal{L}}_{p,q,{\lambda }_1,{\lambda }_2}^{({\alpha }_1,{\alpha }_2,s)}\left(r;u,v\right)-r\left(u,v\right)\right|\le 2\left[{\omega }_1\left(r;\sqrt{{\mathrm{\Psi }}_{p,s,2}(u;{\alpha }_1)}\right)+{\omega }_2\left(r;\sqrt{{\mathrm{\Psi }}_{q,s,2}(v;{\alpha }_2)}\right)\right],$$

where ${\mathrm{\Psi }}_{p,s,2}(u;{\alpha }_1)$ and ${\mathrm{\Psi }}_{q,s,2}(v;{\alpha }_2)$ are defined as in Theorem 3.

Proof. 

By the definition of the partial modulus of the continuity of $r(u,v)$ and the Cauchy–Schwartz inequality, we have:

$$\begin{array}{ccc}& & |{\mathcal{L}}_{p,q,{\lambda }_1,{\lambda }_2}^{({\alpha }_1,{\alpha }_2,s)}\left(r;u,v\right)-r(u,v)|\hfill \\ & & \le {\mathcal{L}}_{p,q,{\lambda }_1,{\lambda }_2}^{({\alpha }_1,{\alpha }_2,s)}\left(\left|r\right(z,t)-r(u,v\left)\right|;u,v\right)\hfill \\ & & \le {\mathcal{L}}_{p,q,{\lambda }_1,{\lambda }_2}^{({\alpha }_1,{\alpha }_2,s)}\left(\left|r\right(z,t)-r(x,t\left)\right|;u,v\right)+{\mathcal{L}}_{p,q,{\lambda }_1,{\lambda }_2}^{({\alpha }_1,{\alpha }_2,s)}\left(\left|r\right(x,t)-r(u,v\left)\right|;u,v\right)\hfill \\ & & \le {\mathcal{L}}_{p,q,{\lambda }_1,{\lambda }_2}^{({\alpha }_1,{\alpha }_2,s)}\left(|{\omega }_1(r;|z-u\left|\right)|;u,v\right)+{\mathcal{L}}_{p,q,{\lambda }_1,{\lambda }_2}^{({\alpha }_1,{\alpha }_2,s)}\left(|{\omega }_2(r;|t-v\left|\right)|;u,v\right)\hfill \\ & & \le {\omega }_1(r,{\mathrm{\Psi }}_{p,s,2}(u;{\alpha }_1))\left[1+\frac{1}{{\mathrm{\Psi }}_{p,s,2}(u;{\alpha }_1)}{\mathcal{L}}_{p,q,{\lambda }_1,{\lambda }_2}^{({\alpha }_1,{\alpha }_2,s)}\left(|z-u|;u,v\right)\right]\hfill \\ & & +{\omega }_2(r,{\mathrm{\Psi }}_{q,s,2}(v;{\alpha }_2))\left[1+\frac{1}{{\mathrm{\Psi }}_{q,s,2}(v;{\alpha }_2)}{\mathcal{L}}_{p,q,{\lambda }_1,{\lambda }_2}^{({\alpha }_1,{\alpha }_2,s)}\left(|t-v|;u,v\right)\right]\hfill \\ & & \le {\omega }_1(r,\sqrt{{\mathrm{\Psi }}_{p,s,2}(u;{\alpha }_1)})\left[1+\frac{1}{\sqrt{{\mathrm{\Psi }}_{p,s,2}(u;{\alpha }_1)}}{\left({\mathcal{L}}_{p,q,{\lambda }_1,{\lambda }_2}^{({\alpha }_1,{\alpha }_2,s)}\left({(z-u)}^2;u,v\right)\right)}^{1/2}\right]\hfill \\ & & +{\omega }_2(r,\sqrt{{\mathrm{\Psi }}_{q,s,2}(v;{\alpha }_2)})\left[1+\frac{1}{\sqrt{{\mathrm{\Psi }}_{q,s,2}(v;{\alpha }_2)}}{\left({\mathcal{L}}_{p,q,{\lambda }_1,{\lambda }_2}^{({\alpha }_1,{\alpha }_2,s)}\left({(t-v)}^2;u,v\right)\right)}^{1/2}\right].\hfill \end{array}$$

Choosing ${\mathrm{\Psi }}_{p,s,2}(u;{\alpha }_1)$ and ${\mathrm{\Psi }}_{q,s,2}(v;{\alpha }_2)$ as defined in Theorem 3, we complete the proof. □

Theorem 7.

Let $r(u,v)\in C^1\left({\mathcal{I}}_{ab}\right)$, then

$$\begin{array}{cc}\hfill & |{\mathcal{L}}_{p,q,{\lambda }_1,{\lambda }_2}^{({\alpha }_1,{\alpha }_2,s)}\left(r;u,v\right)-r(u,v)|\hfill \\ \hfill & \le \sqrt{{\mathrm{\Psi }}_{p,s,2}(u;{\alpha }_1)}\Vert r_u{(u,v)\Vert }_{C\left({\mathcal{I}}_{ab}\right)}+\sqrt{{\mathrm{\Psi }}_{q,s,2}(v;{\alpha }_2)}{\Vert r_v(u,v)\Vert }_{C\left({\mathcal{I}}_{ab}\right)},\hfill \end{array}$$

where ${\mathrm{\Psi }}_{p,s,2}(u;{\alpha }_1)$ and ${\mathrm{\Psi }}_{q,s,2}(v;{\alpha }_2)$ are defined as in Theorem 3.

Proof. 

The following equality holds:

$$\begin{array}{c}\hfill r\left(t\right)-r\left(z\right)={\int }_u^t{r}_u(u,z)du+{\int }_v^z{r}_v(u,v)du\end{array}$$

for $(z,t)\in {\mathcal{I}}_{ab}$. Applying the defined operators on both sides of the last equality, we have:

$$\begin{array}{ccc}& & |{\mathcal{L}}_{p,q,{\lambda }_1,{\lambda }_2}^{({\alpha }_1,{\alpha }_2,s)}\left(r;u,v\right)-r(u,v)|\hfill \\ & & \le {\mathcal{L}}_{p,q,{\lambda }_1,{\lambda }_2}^{({\alpha }_1,{\alpha }_2,s)}\left(\left|{\int }_u^t{r}_u(u,z)du\right|;u,v\right)+{\mathcal{L}}_{p,q,{\lambda }_1,{\lambda }_2}^{({\alpha }_1,{\alpha }_2,s)}\left(\left|{\int }_v^s{r}_v(u,v)du\right|;u,v\right).\hfill \end{array}$$

By the help of the following relations:

$$\begin{array}{ccc}\hfill \left|{\int }_u^t{r}_u(u,z)du\right|& \le & \Vert r_u{(u,v)\Vert }_{C\left({\mathcal{I}}_{ab}\right)}|z-u|;\hfill \\ \hfill \left|{\int }_v^z{r}_v(u,v)du\right|& \le & \Vert r_v{(u,v)\Vert }_{C\left({\mathcal{I}}_{ab}\right)}|t-v|,\hfill \end{array}$$

we have:

$$\begin{array}{ccc}& & |{\mathcal{L}}_{p,q,{\lambda }_1,{\lambda }_2}^{({\alpha }_1,{\alpha }_2,s)}\left(r;u,v\right)-r(u,v)|\hfill \\ & & \le \Vert r_u{(u,v)\Vert }_{C\left({\mathcal{I}}_{ab}\right)}{\mathcal{L}}_{p,q,{\lambda }_1,{\lambda }_2}^{({\alpha }_1,{\alpha }_2,s)}\left(|z-u|;u,v\right)+{\Vert r_v(u,v)\Vert }_{C\left({\mathcal{I}}_{ab}\right)}{\mathcal{L}}_{p,q,{\lambda }_1,{\lambda }_2}^{({\alpha }_1,{\alpha }_2,s)}\left(|t-v|;u,v\right).\hfill \end{array}$$

Using the Cauchy–Schwartz inequality, we have:

$$\begin{array}{ccc}& & |{\mathcal{L}}_{p,q,{\lambda }_1,{\lambda }_2}^{({\alpha }_1,{\alpha }_2,s)}\left(r;u,v\right)-r(u,v)|\hfill \\ & & \le \Vert r_u{(u,v)\Vert }_{C\left({\mathcal{I}}_{ab}\right)}{\left\{{\mathcal{L}}_{p,q,{\lambda }_1,{\lambda }_2}^{({\alpha }_1,{\alpha }_2,s)}\left({(z-u)}^2;u,v\right)\right\}}^{1/2}{\left\{{\mathcal{L}}_{p,q,{\lambda }_1,{\lambda }_2}^{({\alpha }_1,{\alpha }_2,s)}\left(1;u,v\right)\right\}}^{1/2}\hfill \\ & & +\Vert r_v{(u,v)\Vert }_{C\left({\mathcal{I}}_{ab}\right)}{\left\{{\mathcal{L}}_{p,q,{\lambda }_1,{\lambda }_2}^{({\alpha }_1,{\alpha }_2,s)}\left({(t-v)}^2;u,v\right)\right\}}^{1/2}{\left\{{\mathcal{L}}_{p,q,{\lambda }_1,{\lambda }_2}^{({\alpha }_1,{\alpha }_2,s)}\left(1;u,v\right)\right\}}^{1/2}.\hfill \end{array}$$

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5. Graphical Results

In this section, we give three illustrative examples using Mathematica to verify the convergence behavior and accuracy of the studied univariate and bivariate operators for different functions. In Figure 1, Figure 2, Figure 3 and Figure 4, the graphs of functions that are given in the examples and the graphs of the blending univariate and bivariate operators are shown for different $p,$ $q,$ $\alpha ,$ $\lambda$, and s parameters. We also studied the behavior of univariate ${\tilde{b}}^{\alpha ,m}(\lambda ;z)$ and bivariate ${\tilde{b}}_{p,q}^{{\alpha }_1,m}({\lambda }_1;u){\tilde{b}}_{p,q}^{{\alpha }_2,p}({\lambda }_2;v)$ polynomials. The behavior of univariate ${\tilde{b}}^{\alpha ,m}(\lambda ;z)$ and bivariate ${\tilde{b}}_{p,q}^{{\alpha }_1,m}({\lambda }_1;u){\tilde{b}}_{p,q}^{{\alpha }_2,p}({\lambda }_2;v)$ polynomials is demonstrated in Figure 5 and Figure 6, respectively.

Figure 1. Approximation and comparison of univariate operators when shape parameters $\alpha$ and $\lambda$ change. (a) Approximation: $\alpha =0.9,s=3$. (b) Approximation: $\lambda =1,s=3$.

Figure 2. Approximation and error of approximation of operators when shape parameter $\alpha$ changes. (a) Approximation: ${\alpha }_1=0.9,{\alpha }_2=0.1$. (b) Error of approximation: ${\alpha }_1=0.9,{\alpha }_2=0.1$. (c) Approximation: ${\alpha }_1=0.5,{\alpha }_2=0.5$. (d) Error of approximation: ${\alpha }_1=0.5,{\alpha }_2=0.5$.

Figure 3. Approximation and error of approximation of operators when both shape parameters $\lambda$ and $\alpha$ change. (a) Approximation: ${\lambda }_1=-1,{\lambda }_2=1$. (b) Error of approximation: ${\lambda }_1=-1,{\lambda }_2=1$. (c) Approximation: ${\lambda }_1=0.5,{\lambda }_2=-0.5$. (d) Error of approximation: ${\lambda }_1=0.5,{\lambda }_2=-0.5$.

Figure 4. Approximation and error of approximation of operators when shape parameter $\alpha$ changes. (a) Approximation: ${\alpha }_1=0.9,{\alpha }_2=0.1$. (b) Error of approximation: ${\alpha }_1=0.9,{\alpha }_2=0.1$. (c) Approximation: ${\alpha }_1=0.5,{\alpha }_2=0.5$. (d) Error of approximation: ${\alpha }_1=0.5,{\alpha }_2=0.5$.

Figure 5. Approximation of operators when shape parameter $\alpha$ changes (s = 3).

Figure 6. Approximation of operators when shape parameter $\alpha$ changes (s = 3).

Example 1.

We first took the function $r\left(u\right)=exp\left(u^3\right)sin\left(\pi u^8\right)$ into account with $p=10$ to show the convergence of univariate ${\mathcal{L}}_{p,\lambda }^{(\alpha ,s)}$ operators for different values of α and λ in Figure 1. Furthermore, Figure 1 shows the comparison of blending univariate operators. It is clear that the new blending operators have a better approximation thanks to the shape parameters α and λ.

Example 2.

We also show the convergence of ${\mathcal{L}}_{p,q,{\lambda }_1,{\lambda }_2}^{({\alpha }_1,{\alpha }_2,s)}$ to the following bivariate function in Figure 2 and Figure 3:

$$r(u,v)=cos\left(\pi \left(3u^3+1\right)\right)cos\left(\pi \left(3v^3+1\right)\right).$$

Example 3.

We finally used the bivariate function:

$$r(u,v)=\frac{(\pi sin\left(3u^{20}\right)+2u+3)((1-2v)sin\left(\pi v\right))}{(u^3+3)(v^2-3)}$$

in Figure 4 to verify the approximation of bivariate ${\mathcal{L}}_{p,q,{\lambda }_1,{\lambda }_2}^{({\alpha }_1,{\alpha }_2,s)}$ operators with $p=q=20$ and $s=3.$